In [1]:
from IPython.display import Latex
Latex(r"""\begin{eqnarray}
\\ x = \Re(\sqrt{Z_{n}/ |Z_{n}|})
\\ y = \Im(\sqrt{Z_{n}/ |Z_{n}|})*i
\\ dv = |Z_{n} - Z_{0}|
\\ Z_{n+1} = Z_{n} - ( a*x^3 + 3*b*x^2*y + 3*c*x*y^2 + d*y^3 )^{Z_{n}*dv}
\end{eqnarray}""")
Out[1]:
numerical oddity: variation of binomial expansion produces humanoid¶
Sequential iteration of difference equation:
Zr = sqrt(Z/abs(Z));
x = real(Zr);
y = imag(Zr)*i;
d = abs(Z-Z0)
Z = Z - ( p(1)*x^3 + 3*p(2)*x^2*y + 3*p(3)*x*y^2 + p(4)*y^3 )^(Z*d);
with parameter set:
p = [1.92846051108342, 2.27919841968635, 3.37327534248407, 2.17984103218476]
on the complex plane produces a torso and head like figure.In [2]:
import warnings
warnings.filterwarnings('ignore')
import os
import sys
import numpy as np
import time
from IPython.display import display
sys.path.insert(1, '../src');
import z_plane as zp
import graphic_utility as gu;
import itergataters as ig
import numcolorpy as ncp
In [3]:
def T_Spake_Z(Z, p, Z0=None, ET=None):
""" Z = T_Spake_Z(Z, p)
Args:
Z: a real or complex number
p: a real of complex number
Returns:
Z: the result (complex)
"""
d = np.abs(Z-Z0)
Zxy = np.sqrt(Z/np.abs(Z))
x = np.real(Zxy)
y = np.imag(Zxy)*1j
Z = Z - ( p[0]*x**3 + 3*p[1]*x**2*y + 3*p[2]*x*y**2 + p[3]*y**3 )**(Z*d)
return Z
In [4]:
# -- define parameters iterate the above equation --
# -- machine with 8 cores --
par_set = {'n_rows': 800, 'n_cols': 600}
par_set['center_point'] = 1/3 + 0.0j
par_set['theta'] = np.pi/2
par_set['zoom'] = 1/3
par_set['it_max'] = 38
par_set['max_d'] = 10 / par_set['zoom']
par_set['dir_path'] = os.getcwd()
list_tuple = [(T_Spake_Z, ([1.92846051108342, 2.27919841968635, 3.37327534248407, 2.17984103218476]))]
t0 = time.time()
ET, Z, Z0 = ig.get_primitives(list_tuple, par_set)
tt = time.time() - t0
print(tt, '\t total time')
In [5]:
# Escape Time Graphical norm - graphically easier data
t0 = time.time()
Zd, Zr, ETn = ncp.etg_norm(Z0, Z, ET)
print('converstion time =\t', time.time() - t0)
In [6]:
# Escape Time: darker escaped sooner
t0 = time.time()
ETd = ncp.mat_to_gray(ET)
print('coloring time =\t',time.time() - t0)
display(ETd)
In [7]:
# Distance at Escape Time: lighter traveled further
t0 = time.time()
D = ncp.mat_to_gray(Zd)
print('coloring time =\t',time.time() - t0)
display(D)
In [8]:
# Rotation at Escape Time: lighter rotated more
t0 = time.time()
R = ncp.mat_to_gray(Zr)
print('coloring time =\t',time.time() - t0)
display(R)
In [9]:
# Rotation > Hue, Distance > Saturation, Escape Time > Value (intensity-brightness) (muted)
H = ncp.graphic_norm(Zd)
S = ncp.graphic_norm(ET)
V = ncp.graphic_norm(Zr)
Hue_width = 1.0
H_min = 0.0
H_max = H_min + Hue_width
S_max = 1.0
S_min = 0.0
V_max = 1.0
V_min = 0.3
t0 = time.time()
Ihsv = ncp.normat_hsv_intrgb(H, S, V, H_max, H_min, S_max, S_min, V_max, V_min)
print('coloring time:\t',time.time() - t0)
display(Ihsv)
In [10]:
# Rotation > Hue, Escape Time > Saturation, Distance > Value (no significantly muted)
H = 1 - ncp.graphic_norm(Zd)
S = 1 - ncp.graphic_norm(ET)
V = ncp.graphic_norm(Zr)
t0 = time.time()
I = ncp.normat_hsv_intrgb(H, S, V, H_max=1.0, H_min=0.0, S_max=1.0, S_min=0.0, V_max=1.0, V_min=0.3)
print('coloring time:\t',time.time() - t0)
display(I)
In [34]:
H = ncp.range_norm(1 - ETn, lo=0.45, hi=1.0)
S = ncp.range_norm(Zd, lo=0.05, hi=0.1)
V = ncp.range_norm(Zr, lo=0.2, hi=0.8)
t0 = time.time()
Ihsv = ncp.rgb_2_hsv_mat(H, S, V)
print('coloring time:\t',time.time() - t0)
display(Ihsv)
In [34]:
# smaller -> view individual escape time sets
par_set = {'n_rows': 200, 'n_cols': 200}
par_set['center_point'] = 1/3 + 0.0j
par_set['theta'] = np.pi/2
par_set['zoom'] = 1/3
par_set['it_max'] = 64
par_set['max_d'] = 10 / par_set['zoom']
par_set['dir_path'] = os.getcwd()
list_tuple = [(T_Spake_Z, ([1.92846051108342, 2.27919841968635, 3.37327534248407, 2.17984103218476]))]
t0 = time.time()
ET_sm, Z_sm, Z0_sm = ig.get_primitives(list_tuple, par_set)
print(time.time() - t0, '\t total time')
In [38]:
# view smaller - individual escape time starting points
for t in range(2,8):
print('ET =\t',t)
I = np.ones(ET_sm.shape)
I[ET_sm == t] = 0
display(ncp.mat_to_gray(I))
I = np.ones(ET_sm.shape)
I[ET_sm > 7] = 0
display(ncp.mat_to_gray(I))
In [36]:
# view smaller - individual escape time frequency
for k in range(0,int(ET_sm.max())):
print(k, (ET_sm == k).sum())
print('\nHow many never escaped:\n>',(ET_sm > k).sum())
In [37]:
# get the list of unescaped starting points and look for orbit points
Z_overs = Z0[ET_sm == ET_sm.max()]
v1 = Z_overs[0]
d = '%0.2f'%(np.abs(v1))
theta = '%0.1f'%(180*np.arctan2(np.imag(v1), np.real(v1))/np.pi)
print('Unescaped Vector:\n\tV = ', d, theta, 'degrees\n')
print('%9d'%Z_overs.size, 'total unescaped points\n')
print('%9s'%('points'), 'near V', ' (plane units)')
for denom0 in range(1,12):
neighbor_distance = np.abs(v1) * 1/denom0
v1_list = Z_overs[np.abs(Z_overs-v1) < neighbor_distance]
print('%9d'%len(v1_list), 'within V/%2d (%0.3f)'%(denom0, neighbor_distance))
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